theorem Th19:
  weight TM c=iC iff for Am st Am is closed & Am is discrete holds
  card Am c= iC
proof
  hereby
    assume weight TM c=iC;
    then for Fm st Fm is open & Fm is Cover of TM ex Gm st Gm c=Fm & Gm is
    Cover of TM & card Gm c=iC by Th18;
    hence for Am st Am is closed & Am is discrete holds card Am c=iC by Th13;
  end;
  assume for Am st Am is closed & Am is discrete holds card Am c=iC;
  then for Am st Am is discrete holds card Am c=iC by Th14;
  then
  for Fm st Fm is open & not{} in Fm & for Am,Bm st Am in Fm & Bm in Fm &
  Am<>Bm holds Am misses Bm holds card Fm c=iC by Th15;
  then density TM c=iC by Lm6;
  hence thesis by Lm7;
end;
